On the continuity of left translations in the LUC-compactification

Consider the continuity of left translations in the LUC-compactification G LUC of a locally compact group G. For every X ⊆ G , let κ ( X ) be the minimal cardinality of a compact covering of X in G. Let U ( G ) be the points in G LUC that are not in the closure of any X ⊆ G with κ ( X ) < κ ( G ) . We show that the points at which no left translation in U ( G ) is continuous are dense in U ( G ) . This result is a generalization of a theorem by van Douwen concerning discrete groups. We obtain a new proof for the fact that the topological center of G LUC ∖ G is empty.